I have received a letter from Germany on the increase of the elephant, in which a learned Professor arrives at a totally different result from that of Mr. Garbett,1 both of which differ from that of your Correspondent 'Ponderer.' Hence you may perhaps think it worth while to publish a rule by which my son, Mr. George Darwin,2 finds that the product for any number of generations may easily be calculated: —

"The supposition is that each pair of elephants begins to breed when aged 30, breeds at 60, and again, for the last time, at 90, and dies when aged 100, bringing forth a pair at each birth. We start, then, in the year 0 with a pair of elephants, aged 30. They produce a pair in the year 0, a pair in the year 30, a pair in the year 60, and die in the year 70. In the year 60, then, there will be the following pairs alive, viz.: one aged 90, one aged 60, two aged 30, four aged 0. The last three sets are the only ones which will breed in the year 90. At each breeding a pair produces a pair, so that the number of pairs produced in the year 90 will be the sum of the three numbers 1, 2, 4, i.e. 7.Henceforward, at each period, there will be sets of pairs, aged 30, 60, 90 respectively, which breed. These sets will consist of the pairs born at the three preceding periods respectively. Thus the number of pairs born at any period will be the sum of the three preceding numbers in the series, which gives the number of births at each period; and because the first three terms of this series are 1, 2, 4, therefore the series is 1, 2, 4, 7, 13, 24, 44, &c. These are the numbers given by 'Ponderer.' At any period, the whole number of pairs of elephants consists of the young elephants together with the three sets of parents; but since the sum of the three sets of parents is equal in number to the number of young ones, therefore the whole number of pairs is twice the number of young ones, and therefore the whole number of elephants at this period (and for ten years onwards) is four times the corresponding number in the series. In order to obtain the general term of the series, it is necessary to solve an easy equation by the Calculus of Finite Differences."3

CHARLES DARWIN.

1 Edward Lacy Garbett (1817-1887), clergyman and writer. Garbett 1869 claimed there would be 2,400,000 elephants after 500 years and 50,000,000 after six. See Correspondence vol. 17.

2 George Howard Darwin (1845-1912), Darwin's fifth child.

3 See Natural selectionp. 177. There is a sheet of computations on the elephant problem in the Darwin Archive (CUL-DAR46.1.35-36) by George Darwin. One set calculates that based on four sets of young (one pair per set) produced by parents 15,111,870 offspring would be alive after twenty-five generations. The other set calculated the same progression series given by 'Ponderer' and resulted in 5,111,514. Barrett 1977 2: 158 concluded: 'Thus, when writing the Origin of Species in 1858-9, Darwin seems inadvertently to have copied the conditions of the problem from one sheet, and the answer from the second.' In the 6th and final edition of the Origin in 1872, p. 51 Darwin wrote:

The elephant is reckoned the slowest breeder of all known animals, and I have taken some pains to estimate its probable minimum rate of natural increase; it will be safest to assume that it begins breeding when thirty years old, and goes on breeding till ninety years old, bringing forth six young in the interval, and surviving till one hundred years old; if this be so, after a period of from 740 to 750 years there would be nearly nineteen million elephants alive, descended from the first pair.